Algorithm 583: LSQR: Sparse Linear Equations and Least Squares Problems. An iterative method is given for solving Ax = b and min|| Ax - b||2 , where the matrix A is large and sparse. The method is based on the bidiagonalization procedure of Golub and Kahan. It is analytically equivalent to the standard method of conjugate gradients, but possesses more favorable numerical properties. Reliable stopping criteria are derived, along with estimates of standard errors for x and the condition number of A. These are used in the FORTRAN implementation of the method, subroutine LSQR. Numerical tests are described comparing LSQR with several other conjugate-gradient algorithms, indicating that LSQR is the most reliable algorithm when A is ill-conditioned.

This software is also peer reviewed by journal TOMS.

References in zbMATH (referenced in 343 articles , 1 standard article )

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  1. Brezinski, C.: Projection methods for linear systems (1997)
  2. Brezinski, Claude: Projection methods for systems of equations (1997)
  3. Fierro, R. D.; Golub, G. H.; Hansen, P. C.; O’Leary, D. P.: Regularization by truncated total least squares (1997)
  4. Saunders, Michael A.: Computing projections with LSQR (1997)
  5. Wang, Song: An a posteriori error estimate for finite element approximations of a singularly perturbed advection-diffusion problem (1997)
  6. Wright, Stephen J.: Primal-dual interior-point methods (1997)
  7. Zhu, Xiaojun; Lei, Guangtsai; Pan, Guangwen: On application of fast and adaptive periodic Battle-Lemarie wavelets for modeling of multiple lossy transmission lines (1997)
  8. Lukšan, L.: Hybrid methods for large sparse nonlinear least squares (1996)
  9. Resende, Mauricio G. C.; Pardalos, Panos M.: Interior point algorithms for network flow problems (1996)
  10. Vandenberghe, Lieven; Boyd, Stephen: Semidefinite programming (1996)
  11. Bruaset, Are Magnus: A survey of preconditioned iterative methods (1995)
  12. Cao, P.; Macaskill, C.: Iterative techniques for rough surface scattering problems (1995)
  13. Cullum, Jane K.: Peaks, plateaus, numerical instabilities in a Galerkin minimal residual pair of methods for solving (Ax=b) (1995)
  14. Saunders, Michael A.: Solution of sparse rectangular systems using LSQR and Craig (1995)
  15. Vandenberghe, Lieven; Boyd, Stephen: A primal-dual potential reduction method for problems involving matrix inequalities (1995)
  16. van Gijzen, M. B.: Conjugate gradient-like solution algorithms for the mixed finite element approximation of the biharmonic equation, applied to plate bending problems (1995)
  17. Vuik, C.: New insights in GMRES-like methods with variable preconditioners (1995)
  18. Willemans, Karin; Dierckx, Paul: Nonnegative surface fitting with Powell-Sabin splines (1995)
  19. Barker, V. A.; Nielsen, Bo Friis: Using program package NSPCG to analyze the trunk reservation service protection method (1994)
  20. Björck, Åke; Grimme, Eric; van Dooren, Paul: An implicit shift bidiagonalization algorithm for ill-posed systems (1994)

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